Linear Independence of Functions. Wronskian

The functions \({y_1}\left( x \right),{y_2}\left( x \right), \ldots ,{y_n}\left( x \right)\) are called linearly dependent on the interval \(\left[ {a,b} \right],\) if there are constants \({\alpha _1},{\alpha _2}, \ldots ,{\alpha _n},\) not all zero, such that for all values of \(x\) from this interval, the identity

For the case of two functions, the linear independence criterion can be written in a simpler form: The functions \({y_1}\left( x \right),\) \({y_2}\left( x \right)\) are linearly independent on the interval \(\left[ {a,b} \right],\) if their quotient in this segment is not identically equal to a constant:

is called the Wronski determinant or Wronskian for this system of functions.

Wronskian Test.

If the system of functions \({y_1}\left( x \right),\) \({y_2}\left( x \right), \ldots ,\) \({y_n}\left( x \right)\) is linearly dependent on the interval \(\left[ {a,b} \right],\) then its Wronskian vanishes on this interval.

It follows from here that if the Wronskian is nonzero at least at one point in the interval \(\left[ {a,b} \right],\) then the functions \({y_1}\left( x \right),\) \({y_2}\left( x \right), \ldots ,\) \({y_n}\left( x \right)\) are linearly independent. This property of the Wronskian allows to determine whether the solutions of a homogeneous differential equation are linearly independent.

Fundamental System of Solutions

A set of two linearly independent particular solutions of a linear homogeneous second order differential equation forms its fundamental system of solutions.

If \({y_1}\left( x \right),{y_2}\left( x \right)\) is a fundamental system of solutions, then the general solution of the second order equation is represented as

Note that for a given fundamental system of solutions \({y_1}\left( x \right),\) \({y_2}\left( x \right)\) we can construct the corresponding homogeneous differential equation. For the case of a second order equation, it is expressed in terms of the determinant:

Liouville’s Formula

Thus, as noted above, the general solution of a homogeneous second order differential equation is a linear combination of two linearly independent particular solutions \({y_1}\left( x \right),\) \({y_2}\left( x \right)\) of this equation.

Obviously, the particular solutions depend on the coefficients of the differential equation. The Liouville formula establishes a connection between the Wronskian \(W\left( x \right),\) constructed on the basis of particular solutions \({y_1}\left( x \right),\) \({y_2}\left( x \right),\) and the coefficient \({a_1}\left( x \right)\) in the differential equation.

Let \(W\left( x \right)\) be the Wronskian of the solutions \({y_1}\left( x \right),\) \( {y_2}\left( x \right)\) of a linear second order homogeneous differential equation

in which the functions \({a_1}\left( x \right)\) and \({a_2}\left( x \right)\) are continuous on the interval \(\left[ {a,b} \right].\) Let the point \({x_0}\) belong to the interval \(\left[ {a,b} \right].\) Then for all \(x \in \left[ {a,b} \right]\) the Liouville formula

Practical methods for solving second order homogeneous equations with variable coefficients

Unfortunately, the general method of finding a particular solution does not exist. Usually this is done by guessing.

If a particular solution \({y_1}\left( x \right) \ne 0\) of the homogeneous linear second order equation is known, the original equation can be converted to a linear first order equation using the substitution \(y = {y_1}\left( x \right)z\left( x \right)\) and the subsequent replacement \(z’\left( x \right) = u.\)

Another way to reduce the order is based on the Liouville formula. In this case, a particular solution \({y_1}\left( x \right)\) must also be known. The relevant examples are given below.